Subgroups — #30772140

Order 17152.

442 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561820	C₂	1	1	yes	maximal, characteristic
2	#22560247	C₂	1	1	yes	maximal, characteristic
3	#22558317	C₂	1	4	yes	maximal, complemented
4	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523979	C₂²	1	2	yes	—
7	#16863360	—	2	1	no	—
8	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
10	#16863161	—	2	1	no	—
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
13	C₂×#107×C₆₇	—	2	1	no	—
14	C₂×#107×C₆₇	—	2	1	no	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	C₄	1	2	yes	—
17	C₂×#107×C₆₇	C₄	1	2	yes	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#11761612	—	2	2	no	—
20	#14523983	—	2	4	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	#14523979	—	2	4	no	—
23	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
25	C₂³×C₄×C₆₇	—	4	1	no	—
26	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	2	4	no	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
40	#107×C₆₇	—	2	2	no	—
41	#107×C₆₇	—	2	2	no	—
42	#107×C₆₇	—	2	2	no	—
43	#107×C₆₇	C₂×C₄	1	4	yes	—
44	#11761612	—	4	2	no	—
45	#11761612	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₆₇⋊C₁₆	—	4	2	no	—
48	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	—	4	2	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	8	4	no	—
55	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	#41	1	4	yes	—
71	C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₄⋊C₄×C₆₇	—	4	2	no	—
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₂³×C₆₇	—	2	2	no	—
76	C₂³×C₆₇	—	2	2	no	—
77	C₂³×C₆₇	—	2	2	no	—
78	C₂³×C₆₇	#102	1	1	yes	characteristic
79	C₂³×C₆₇	#102	1	1	yes	characteristic
80	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	8	4	no	—
90	C₂×C₄×C₆₇	—	8	4	no	—
91	C₂×C₄×C₆₇	—	8	4	no	—
92	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#123	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#137	1	2	yes	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	#117	1	2	yes	—
101	C₈×C₆₇	#123	1	2	yes	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	—	2	8	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	#361	1	2	yes	—
111	C₂²×C₆₇	#362	1	2	yes	—
112	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
113	C₂²×C₆₇	#483	1	2	yes	—
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	8	4	no	—
117	C₄×C₆₇	#381	1	1	yes	characteristic
118	C₄×C₆₇	#419	1	1	yes	characteristic
119	C₄×C₆₇	#480	1	2	yes	—
120	#12871	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	2	no	—
122	C₂×C₆₇	—	2	2	no	—
123	C₂×C₆₇	—	2	2	no	—
124	C₂×C₆₇	#1823	1	1	yes	characteristic
125	C₂×C₆₇	C₂×#361	1	1	yes	characteristic
126	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
127	C₂²×#107	D₆₇	1	1	yes	characteristic
128	#2112	—	67	1	no	—
129	#1810	—	67	1	no	—
130	#1387	—	67	4	no	—
131	C₆₇	#12871	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	—	2	1	no	—
134	C₂×M₃₂	—	67	2	no	—
135	C₂×#108	—	67	2	no	—
136	C₂×#107	—	2	1	no	—
137	C₂×#107	—	2	1	no	—
138	C₂×#107	—	2	1	no	—
139	C₂×#107	Dic₆₇	1	2	yes	—
140	C₂×#107	Dic₆₇	1	2	yes	—
141	C₂×#107	C₂×D₆₇	1	2	yes	—
142	#407	—	134	1	no	—
143	C₂²⋊C₁₆	—	134	1	no	—
144	C₂²⋊C₁₆	—	134	1	no	—
145	#341	—	134	1	no	—
146	C₂³×C₄	—	4	1	no	—
147	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂×M₁₆	—	134	2	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	4	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	M₃₂	—	134	4	no	—
163	C₂×C₁₆	—	134	2	no	—
164	#108	—	134	4	no	—
165	#107	—	2	2	no	—
166	#107	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	C₂×Dic₆₇	1	4	yes	—
169	C₂²⋊C₈	—	268	1	no	—
170	C₂²⋊C₈	—	268	1	no	—
171	C₂⁴	#11761616	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	8	4	no	—
178	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
180	C₂²×C₄	#11761613	1	1	yes	characteristic
181	C₂²×C₄	#11761613	1	1	yes	characteristic
182	C₂²×C₄	#11761616	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
184	M₁₆	—	268	2	no	—
185	M₁₆	—	268	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	268	2	no	—
192	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
194	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂×C₈	#11761628	1	4	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₁₆	—	268	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	#14523964	1	1	yes	characteristic
206	C₂³	#14523964	1	1	yes	characteristic
207	C₂³	C₂×#11761616	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	#14523964	1	1	yes	characteristic
220	C₂×C₄	#14523964	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
222	C₂×C₄	#14524026	1	2	yes	—
223	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
224	C₂×C₄	#14524074	1	2	yes	—
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	536	2	no	—
228	C₈	#14524002	1	2	yes	—
229	C₈	#14524026	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	#16863209	1	2	yes	—
238	C₂²	#16863212	1	2	yes	—
239	C₂²	C₂×#14523964	1	1	yes	characteristic
240	C₂²	#16863726	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	4	no	—
244	C₄	#16863259	1	1	yes	characteristic
245	C₄	#16863405	1	1	yes	characteristic
246	C₄	#16863709	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	#22560295	1	1	yes	characteristic
251	C₂	C₂×#16863209	1	1	yes	characteristic
252	C₂	C₂×#16863212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#107×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#107	D₆₇
hypocenter	C₆₇	#12871
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #22558317	C₂	semidirect
131. C₆₇	#12871	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22561820	2
#22560247	2
#22558317	2
C₂²×#107×C₆₇	2
#12871	67